Theorem 2

a) If I is a finite set of n elements, then $\varphi_I$ induces a bicontinuous isomorphism of $A[[S_1, \ldots, S_n]]$ onto $A[[X]]^{\text{sym}}$.

b) If I is infinite, $\varphi_I$ is a bicontinuous isomorphism of $A[[S]]$ onto $A[[X]]^{\text{sym}}$.

In case a) we put $B = A[[S_1, \ldots, S_n]]$ and equip this algebra with the topology induced by that of $A[[S]]$; we also denote by $\psi_I$ the restriction of $\varphi_I$ to B. In case b) we put $B = A[[S]]$ and $\psi_I = \varphi_I$. We shall equip the polynomial algebra $A[S]$ with the graduation of type N for which $S_k$ is of weight k for every integer $k \geq 1$.